Question: The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $29.3$ years; the standard deviation is $5.9$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living between $41.1$ and $47$ years.
Solution: $29.3$ $23.4$ $35.2$ $17.5$ $41.1$ $11.6$ $47$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $29.3$ years. We know the standard deviation is $5.9$ years, so one standard deviation below the mean is $23.4$ years and one standard deviation above the mean is $35.2$ years. Two standard deviations below the mean is $17.5$ years and two standard deviations above the mean is $41.1$ years. Three standard deviations below the mean is $11.6$ years and three standard deviations above the mean is $47$ years. We are interested in the probability of a snake living between $41.1$ and $47$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the snakes will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the snakes will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of snakes between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular snake living between $41.1$ and $47$ years is $\color{orange}{2.35\%}$.